\documentclass[10pt]{article}
\usepackage{bm}% bold math
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{color}
\DeclareMathOperator{\sgn}{sgn}
\renewcommand{\thefootnote}{\alph{footnote}}

\begin{document}

\subsection{Ideal Gas Model}

From (McQuarrie, Statistical mechanics, 2000) the ideal gas method (IGM) for calculating an absolute free energy is outlined below.
\begin{equation}
	G  = H - TS
\end{equation}
\begin{equation}
	H = U + RT
\end{equation}
\begin{equation}
	U = E_\text{pot} + E_\text{ZPE} + E_\text{trns} + E_\text{rot} + E_\text{vib} \end{equation}
\begin{equation}
	S = S_\text{trns} + S_\text{rot} + S_\text{vib} + S_\text{elec}
\end{equation}

where $T$ is temperature, $R$ the ideal gas constant and $S_\text{elec}$ is taken to be zero for all molecules. The internal energy components are then

\begin{equation}
	E_\text{ZPE} = \frac{N_a}{2}\sum_i h \nu_i
\end{equation}
\begin{equation}
	E_\text{trns} = \frac{3}{2}RT
\end{equation}
\begin{equation}
	E_\text{rot} =
	\begin{cases}
		0 &\quad \text{if  } N = 1 \\
		RT &\quad \text{if linear} \\
		\frac{3}{2} RT &\quad \text{otherwise}
	\end{cases}
\end{equation}
\begin{equation}
	E_\text{vib} = R \sum_i \frac{\theta_i}{e^{\theta_i / T} - 1} \quad ,\quad \theta_i = h\nu_i / k_B
\end{equation}

where $N_a$ is Avogadro's's constant, $N$ is the number of atoms in the molecule, $k_B$ Boltzmann's constant, $ \nu_i$ the $i$-th harmonic frequency and $h$ is Planks constant. The entropic components are

\begin{equation}
	S_\text{trns} = R  \ln(q_\text{trns}) + \frac{5}{2}R
\end{equation}
\begin{equation}
	S_\text{rot} = \begin{cases}
		0 &\quad \text{if  } N = 1 \\
		R  \ln(q_\text{rot}) + R &\quad \text{if linear} \\
		R  \ln(q_\text{rot}) + \frac{3}{2}R &\quad \text{otherwise}
	\end{cases}
\end{equation}
\begin{equation}
	S_\text{vib}^\text{HO} = R \sum_i \frac{\theta_i}{T(e^{\theta_i / T} - 1)} - \ln(1 - e^{-\theta_i / T})
\end{equation}
\begin{equation}
	q_\text{trans} = {\Big (} \frac{2\pi m k_B T}{h^2} {\Big )}^{3/2}  V_\text{eff} \quad , \quad V_\text{eff} =  \begin{cases}
		k_B T / p^{\circ} \quad&\text{if 1 atm standard state} \\
		1 / c^\circ N_a \quad&\text{if 1 M standard state}
	\end{cases}
\end{equation}
\begin{equation}
	q_\text{rot} = \frac{T^{3/2}}{\sigma_r} \sqrt{\frac{\pi}{\omega_r}} \quad,\quad \omega_r = \prod_{k} \frac{h^2}{8 \pi^2 k_B I_k}
\end{equation}

where $q$ are molecular partition functions, $p^{\circ}$ is the standard pressure (1 atm) and $c^\circ$ the standard concentration (1 mol dm$^{-3}$), $\sigma_r$ is the rotational symmetry number for the molecule and $I_k$ a diagonal element of the moment of inertia matrix.
\\\\
Due to the vibrational entropy contribution being overestimated for low frequency modes Thrular proposed a correction, which instead of summing over frequencies in $S_\text{vib}^\text{HO}$ does so over $\max(\nu_\text{thresh},\; \nu_i)$ to shift all low frequencies to a threshold value (\emph{J. Phys. Chem. B} 2011, {\bfseries{115}}, 14556). An alternative method from Grimme (\emph{Chem. Eur. J.}, 2012, {\bfseries{18}}, 9955) uses an interpolation between a harmonic oscillator and rigid rotor to scale down the contribution from the low frequency modes as
\begin{equation}
	S_\text{vib}^\text{Grimme} = \sum_i w_i S_ \text{vib}^\text{HO}(i) + (1-w_i) {\Big (} R\ln {\Big (} \sqrt{\frac{8 \pi^3 \mu_i' k_B T}{h^2}} {\Big )} + \frac{R}{2} {\Big )}
\end{equation}
\begin{eqnarray}
	\mu_i' = \frac{\mu_i \bar{B}}{\mu_i + \bar{B}} \quad,\quad \mu_i = \frac{h}{8\pi^2 \nu_i} \quad,\quad \bar{B} = \text{Tr}[I] / 3
\end{eqnarray}
\begin{equation}
	w_i = \frac{1}{1 + (\omega_0/ \nu_i)^\alpha}
\end{equation}
where $\omega_0$ and $\alpha$ are adjustable parameters.







\end{document}